| L(s) = 1 | + 2-s − 7·4-s + 5·5-s − 6·7-s − 15·8-s + 5·10-s − 47·11-s − 5·13-s − 6·14-s + 41·16-s − 131·17-s − 56·19-s − 35·20-s − 47·22-s + 3·23-s + 25·25-s − 5·26-s + 42·28-s − 157·29-s + 225·31-s + 161·32-s − 131·34-s − 30·35-s − 70·37-s − 56·38-s − 75·40-s + 140·41-s + ⋯ |
| L(s) = 1 | + 0.353·2-s − 7/8·4-s + 0.447·5-s − 0.323·7-s − 0.662·8-s + 0.158·10-s − 1.28·11-s − 0.106·13-s − 0.114·14-s + 0.640·16-s − 1.86·17-s − 0.676·19-s − 0.391·20-s − 0.455·22-s + 0.0271·23-s + 1/5·25-s − 0.0377·26-s + 0.283·28-s − 1.00·29-s + 1.30·31-s + 0.889·32-s − 0.660·34-s − 0.144·35-s − 0.311·37-s − 0.239·38-s − 0.296·40-s + 0.533·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 135 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 135 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 - p T \) |
| good | 2 | \( 1 - T + p^{3} T^{2} \) |
| 7 | \( 1 + 6 T + p^{3} T^{2} \) |
| 11 | \( 1 + 47 T + p^{3} T^{2} \) |
| 13 | \( 1 + 5 T + p^{3} T^{2} \) |
| 17 | \( 1 + 131 T + p^{3} T^{2} \) |
| 19 | \( 1 + 56 T + p^{3} T^{2} \) |
| 23 | \( 1 - 3 T + p^{3} T^{2} \) |
| 29 | \( 1 + 157 T + p^{3} T^{2} \) |
| 31 | \( 1 - 225 T + p^{3} T^{2} \) |
| 37 | \( 1 + 70 T + p^{3} T^{2} \) |
| 41 | \( 1 - 140 T + p^{3} T^{2} \) |
| 43 | \( 1 - 397 T + p^{3} T^{2} \) |
| 47 | \( 1 + 347 T + p^{3} T^{2} \) |
| 53 | \( 1 - 4 T + p^{3} T^{2} \) |
| 59 | \( 1 - 748 T + p^{3} T^{2} \) |
| 61 | \( 1 + 338 T + p^{3} T^{2} \) |
| 67 | \( 1 - 492 T + p^{3} T^{2} \) |
| 71 | \( 1 - 32 T + p^{3} T^{2} \) |
| 73 | \( 1 - 970 T + p^{3} T^{2} \) |
| 79 | \( 1 + 1257 T + p^{3} T^{2} \) |
| 83 | \( 1 + 102 T + p^{3} T^{2} \) |
| 89 | \( 1 + 1488 T + p^{3} T^{2} \) |
| 97 | \( 1 - 974 T + p^{3} T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−12.77732423259932501718338943451, −11.17684380951355680080293206482, −10.10833635728261630529661204567, −9.124161274570911954009567994425, −8.132703514134974010640967193916, −6.54882856740379749779071491816, −5.34358094938513260771629321003, −4.25832988831698386516179103786, −2.56286497032969400432426342053, 0,
2.56286497032969400432426342053, 4.25832988831698386516179103786, 5.34358094938513260771629321003, 6.54882856740379749779071491816, 8.132703514134974010640967193916, 9.124161274570911954009567994425, 10.10833635728261630529661204567, 11.17684380951355680080293206482, 12.77732423259932501718338943451
